Gear tooth form

ABSTRACT

Improved form for the profiles of a pair of mating gears having conjugate gear teeth provides substantially constant relative curvature at each point of contact. A differential equation, as well as a more easily solvable approximate solution for the differential equation is given to enable the path of contact of the pair of gears to be determined. The improved tooth form incorporates the major advantages of involute gearing but is able to take substantially higher loading, requires no undercut for gears having a low number of teeth, and provides more intimate contact. The tooth form may be cut with standard machines but has especial utility in powder metal gears since tooth strength can often be increased sufficiently to permit powder metal gears to replace cut gears.

United States Patent App]. No. Filed Patented Assignee GEAR TOOTH FORM 7Claims, 4 Drawing Figs.

U.S. Cl

Int. Cl Field of Search References Cited UNITED STATES PATENTS 2/1936Hill 8/1937 Hill 2/ 1941 Wildhaber 74/462 Fl6h 55/06 74/462 3,251,2365/1966 Wildhaber 3,371,552 3/1968 Soper ABSTRACT: Improved form for theprofiles of a pair of mating gears having conjugate gear teeth providessubstantially constant relative curvature at each point of contact. Adifferential equation, as well as a more easily solvable approximatesolution for the differential equation is given to enable the path ofcontact of the pair of gears to be determined. The improved tooth formincorporates the major advantages of involute gearing but is able totake substantially higher loading, requires no undercut for gears havinga low number of teeth, and provides more intimate contact. The toothform may be cut with standard machines but has especial utility inpowder metal gears since tooth strength can often be increasedsufficiently to permit powder metal gears to replace cut gears.

GEAR 'roorrr FORM BACKGROUND OF THE INVENTION 1. Field of the InventionThis invention relates to gearing and particularly to the shape of thetooth profile.

2. Description of the Prior Art The involute curve as a tooth form forspur and helical gears is almost universally used in modern gear design,because of its very real advantages over the infinite number of otherpossible forms. The main advantages are as follows:

1. Simplicity of cutting tools, resulting from the straightsided rackform which is the basis of the involute.

2. lnsensitivity of involute gears to errors in mounting centerdistance. Conjugate action remains theoretically correct in allpositions.

3. Basic simplicity of the mathematics involved in designing involutegear teeth, andreliable rating formulas now available, based on a vastamount of test and field usage data.

Today s involute gears carry far more power with greater reliabilitythan was once thought possible. Improvements in material and lubricationand more precise manufacture, made possible by modern equipment, aremainly responsible. Still the search for more strength goes on, asindicated by continuing test programs at many laboratories. As furtherprogress from these standard approaches becomes more and more difficult,it becomes worthwhile to look for a better gear geometry than theinvolute, as being the only direction in which significant gains can bemade.

Many modifications from the involute curve for gears have been tried inthe past. Slight modifications such as tip relief and crown have'proventheir value and are an essential part of the best modern gear design.More radical departures, such as the Wildhaber-Novikov tooth forms, haveshown some promise under test conditions; however, the loss of thepreviously noted three fundamental advantages of the involute form is aserious disadvantage, and none of these designs has received wideacceptance.

The one principal disadvantage of the involute form is the rapidlydiminishing radius of curvature of the involute curve in the vicinity ofthe base circle. At the base circle the radius of curvature becomeszero. The contact stresses (Hertz stresses) between gear teeth becomelarger as the radii of curvature becomes smaller; in fact, at the basecircle the stress would be theoretically infinite. Hence, involute gearsshould never be designed for contact at or near the base circle. Gooddesign can usually minimize this drawback of the involute geometry;nevertheless, in many highly optimized designs, contact stress is stillthe principal limitation on load capacity.

Gears of the involute type start to lose part of their active profiles(to undercut") whenever the generating tool extends below the pointwhere the path of contact touches the base circle. For 20 pressure anglestandard depth teeth this occurs at 17 teeth and under. The teeth alsobecome weaker in beam strength. To overcome these difiiculties, "longand short addendum systems are often used. This recourse fails when twogears with low numbers of teeth must run together, as in some planetaryarrangements. These modifications also have a detrimental effect when alarge gear tends to overrun, or back-drive its pinion.

That others have considered the possibility of defining gear tooth formswhich retain in substance the main advantages of the involute, but avoidthe adverse tooth curvature condition, is evidenced by US. Pat. Nos.2,128,815 to James Guest and 3,251,236 to Ernest Wildhaber. Each ofthese patents show forming the profile of the basic rack as a sine curverather than as a straight line as in the case of an involute.

From elementary gear theory it can be readily understood that the toothprofiles of all gears operating on parallel axes must obey the well!-known conjugate action law: the common normal at all points of contactmust pass through a fixed point on the line of centers, called the pitchpoint. This is a kinematic requirement if one profile is to drive theother at a constant angular speed ratio. It can also-be readilyunderstood that a pair of gear profiles contact each other at differentpositions as the gears rotate. The locus of all-possible contact pointsfor a given pair of profiles is called the path of contact. This is astraight or curved line segment, terminated by the extremities of thegear teeth. The three curves involved in the most fundamental part ofgear design are (l) the profile of gear No. l; (2) the profile of gearNo. 2; and (3) the path of contact.

A basic geometric fact of great significance is, that, given a fixedcenter distance and speed ratio, any one of these curves completelydetennines the other two. For example, if the profile of gear No. l ismade some definite mathematical curve; then the profile of gear No. 2and the path of contact are both uniquely determined. Likewise, if thepath of contact is shown as some given curve, the profiles of both gearsare uniquely determined. Thus it is possible to find mathematicalrelationships between tooth curvatures from given properties of the pathof contact.

SUMMARY [t is an object of this invention to achieve a gear toothprofile which the major advantages of the involute form whileeliminating most of the disadvantages.

It is a further object of this invention to determine a mathematicalequation which will define the path of contact, and thus the toothprofiles, of a pair of mating gears having greatly improved strengthcharacteristics as compared to gears with involute profiles.

It is an additional object of this invention to determine anapproximate, and readily solvable, solution for a difierent equationdefining a path of contact, and thus the tooth profiles, of a pair ofmating gears which'will have greatly improved strength characteristicsas compared to gears having involute profiles.

These and other objects are attained by the present invention whichprovides for the profiles of a pair of mating conjugate gear teeth ashaving substantially constant relative curvature at each point ofcontact. The profile of the basic rack and thus the profiles of themating gears can be determined, assuming the fixed center distancebetween the gears and the speed ratio is known, from the coordinates ofthe path of contact. Once the path of contact is known, it is a simplematter for the basic-rack profile to be determined since equations fordetermining the rack profile from the path of contact are well known tothe gear designer (see pages 15 and 16, Buckingham, Analytical Mechanicsof Gears, McGraw-Hill Book Co. 1949).

For an involute system, the path of contact is a straight line and therelative curvature for each of its mating gears near their base circleapproaches infinity. Since a large relative curvature indicates a greatprobability of surface failure, one can readily understand that aninvolute gear is weak near its base circle. The weakness problem ofinvolute gears is emphasized even further when one considers than forinvolute gears having relatively few teeth (less than about 16) theteeth are undercut near their base circle. It may be seen that at thepitch contact point the relative curvature for all systems depends onlyon the pitch radii and the pitch pressure angle-not on the general shapeof the path of contact. Thus, it would seem that to optimize a geardesign, and keep the Hertz stress from exceeding its value at the pitchpoint, the relative curvature at every contact point should be equal toor less than that at the pitch point. There will be a minimum deviationfrom the involute, especially with respect to insensitivity to changesin center distance, if the tooth form has the property of constantrelative curvature. The reason for trying to achieve a minimum deviationfrom the involute is to retain as many of its desirable properties aspossible. Thus, whether the relative curvature at every point on thepath of contact is slightly greater or slightly less than at the pitchpoint (and variations in both directions are contemplated by thisinvention), the advantages of the involute system with regard toinsensitivity to center distance variations will be substantiallymaintained. Although a slight increase in the relative curvature willslightly increase the Hertz stress as compared to the case of a gearwhere the relative curvature is maintained exactly constant or slightlyless than at the pitch point, the difference would be of relativelylittle significance in the performance of the gear.

By means of a differential equation which will be hereinafter set forth,and whose derivation is not necessary to an understanding of the presentinvention, one can obtain the solution, in terms of the polar coordinatevariables 45 and s, for the path of contact. Knowing the path ofcontact, as previously noted, the gear tooth and rack forms can then bedetermined. Since the differential equation is not linear, the solutionis not easy to come by. However, one may use a procedure ofapproximating a solution by obtaining a series of terms. It has beenfound that by carrying the solution to only three terms it is in manycases possible to achieve a relative curvature which is within l percentof its value at the pitch point at every point on the path of contact.Even on gears having relatively few numbers of teeth, such as eight, ithas been possible to achieve a relative curvature which is within -20percent of its value at the pitch point. A corresponding eighttoothedinvolute gear would be extremely weak since it would be undercut in itsbase and would have a relative curvature at its extremes of contactwhich would be several times its value at the pitch point, leading tohigh Hertz stresses.

The constant relative curvature gear design gives significant advantagesmainly in cases where at least one of the mating gears has a low numberof teeth-say less than 22. There are many such applications, some ofwhich are: final drives of vehicles; high ratio speed reducers;planetaries; radar pedestal drives; pump gears; parallel axis gearing;and in many sintered metal spun gear applications. Gear teeth ofconstant relative curvature never undercut, and have greater beamstrength than involute gear teeth of the same proportions. Furthermore,these advantages are maintained even when the series solution to thedifi'erentiai equation is approximated by being truncated to threeterms. The approximate solution is actually desirable since the relativecurvature diminishes very slightly to each side of the pitch point andresults in the teeth being slightly strengthened in these regions wheresliding action is at a maximum.

Gears having constant (or substantially constant) relative curvature canbe produced by the same processes as involute gears, at a very littleincrease in cost. When the gears are produced from powdered metal bysintering, the increase in production cost-occurring only in making themolds-is negligible, and the gain in strength can be quite substantialdue to the limitations in effective surface hardness obtainable inpowdered metal. The same holds true for plastic gears.

Where the gears are produced by hobbing, the hob is slightly morecomplex, having a curved instead of straight shape. However, suchmodifications are easy to accomplish with present-day hob-makingtechniques.

Where the gear teeth are finished by form grinding, as in many criticalaircraft power trains, the tooth having a constant relative curvaturerequires only a different template, no more difficult to make than theinvolute.

For shaped and shaved gears, the tools at present might involve somedifficulties, which, however, can be overcome with modified equipment.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a diagram showing variousdimensions of a pair of mating gears and illustrating the three basiccurves fundamental to gear design;

FIG. 2 is a diagram showing polar coordinates being used to define thepath of contact of a pair of mating gears;

FIG. 3 is a graph plotting relative curvature versus roll angle frompitch point for eight-tooth gears having involute and constant relativecurvature profiles; and

FIG. 4 is a diagram showing the tooth profile of an eighttooth gearhaving constant relative curvature as compared to an involute gear.

DESCRIPTION OF THE PREFERRED EMBODIMENT Referring to FIG. I, the threecurves fundamental to gear design may be seen. These are: PG, which isthe profile of the first gear indicated generally at l and having acenter at C, and an outside radius R0,; PG, which is the profile of thesecond gear indicated generally at 2 and having a center at C, and anoutside radius R0,; and PC which is the path of contact. Otherdimensions of interest on gears l and 2 are their center-tocenterdistance CD and their pitch radii, R, and R The center-to-centerdistance CD is the distance between the centers C, and C, while thepitch radii R, and R, are the distances between the pitch point and therespective gear centers C, and C As the gears l and 2 rotate, the toothprofile curves PG, and PG, will contact each other at differentpositions and the locus of all successive contact points will determinethe shape of the path of contact PC.

In FIG. 2, a pair of gears l and 2 are illustrated in a difierentposition than in FIG. 1 with a point of contact CP being indicated interms of polar coordinates as being located at a distance r from thepitch point PP and at an angle d: which is also the pressure angle froma horizontal line normal to the line connecting centers C, and C Theradius of curvature of the profile PG, of gear 1 is of a length p, whilethe radius of curvature of profile P6 of gear 2 is of a length p Aspreviously noted (and assuming a fixed center distance CD and speedratio R IRJ any one of the curves PG,, PG: and PC completely determinesthe other two. Thus, if the path of contact PC is shown as some givencurve, the profiles PG, and PG, of gears l and 2 will be uniquelydetermined. Further more, it is possible to find mathematicalrelationships between tooth curvatures or profiles from given propertiesof the path of contact. These mathematical relationships form the basisfor designing gears having teeth of constant relative curvature.

Referring to FIG. 2, the general formulas for tooth curvatures p, and pas well as for relative tooth curvature K, are as follows:

1R, cos 4) 1:1 Pr P1 p2 sin '11,, constant The solution could of coursebe carried farther to obtain more terms, but from a practical standpointadditional terms are unnecessary.

FIG. 3 graphically plots a curve 5 of relative curvature K, versus rollangle from the pitch point for an eight-tooth involute gear and asimilar curve 6 for an eight-tooth gear having substantially constantrelative curvature (as determined from the aforementioned approximationequations). It can be readily noted that curve 6 is slightly curved downat its ends. This shape means that the Hertz stress is slightly less atthe beginning and end of the tooth profile PG, a desirable proper tysince the mating teeth are subject to more wear at the ends of theirprofiles where they are undergoing a sliding motion than at the pitchpoint where their relative motion is a rolling one. Looking at curve 5for the involute, it is obvious that the Hertz stress is extremely highat the ends of the tooth profile. Thus, the involute teeth will begreatly weakened at these points. It can also be seen in FIG. 3 thatcurve 6 extends further in a horizontal direction than curve 5, anindication that teeth with constant relative curvature can be made toremain in contact for several more degrees of rotation. Such additionalcontact is very beneficial since it lessens the tooth loading.

In FIG. 4, a comparison between the active profile PG, of an eight-toothgear 10 having constant relative curvature and the active profile PG, ofan eight-tooth involute gear having a base circle 12 is shown. It can bereadily seen that the active profile PG, of the involute tooth is muchshorter and therefore much less able to offer good contact carryoverfrom tooth to tooth.

In designing a gear system utilizing the concept of constant relativecurvature disclosed herein, the designer need merely determine theapproximate solution for a sufficient number of points, such as 25 forexample, on the path of contact to enable the path of contact, and fromit, the tooth profiles, to be determined with the accuracy desired.Where many points are to be detennined it has been found extremelyhelpful to utilize a digital computer.

I claim:

1. A gear set comprising a pair of mating gears having externalconjugate gear teeth and one of said gears having less than 30 teeth,the profile of said gear teeth having substantially constant relativecurvature at all points of contact.

2. A gear set as defined in claim 1 wherein the relative curvatures ofthe gear teeth at all contact points is within percent of its value atthe pitch point.

3. A gear set as defined in claim 1 wherein the relative curvatures ofthe gear teeth at all contact points is within 5 percent of its value atthe pitch point.

4. A gear set as defined in claim 1 wherein the relative curvatures ofthe gear teeth at all contact points is within 1 percent of its value atthe pitch point.

5. A gear set comprising a pair of mating ears having conjugate gearteeth, the path of contact of said pair of gears being a curvesubstantially satisfying the differential equation:

R Rz sin (11 Where:

R pitch radius of gear no. 1,

R pitch radius of gear no. 2,

d), the instantaneous value of the pressure angle at the pitch contactpoint,

s, d: are the polar coordinates of the path of contact,

referred to the pitch point, with also being the instantaneous pressureangle at any contact point,

p radius of curvature of profile of gear no. 1,

p radius of curvature of profile of gear no. 2,

p,= relative radius of curvature,

and by definition:

6. A gear set as defined in claim 5 wherein said path of contactcorresponds to the approximate solution to said differential equationwhich is defined by the equation:

=constant 1 (2a R1 a2 5 sin p R3 7. A gear cutting tool for cutting oneof a set of mating gear teeth, the basic rack form of said cutting toolbeing derived from the equation:

where:

l a sin :11, 1 *3 R2

1. A gear set comprising a pair of mating gears having externalconjugate gear teeth and one of said gears having less than 30 teeth,the profile of said gear teeth having substantially constant relativecurvature at all points of contact.
 2. A gear set as defined in claim 1wherein the relative curvatures of the gear teeth at all contact pointsis within 20 percent of its value at the pitch point.
 3. A gear set asdefined in claim 1 wherein the relative curvatures of the gear teeth atall contact points is within 5 percent of its value at the pitch point.4. A gear set as defined in claim 1 wherein the relative curvatures ofthe gear teeth at all contact points is within 1 percent of its value atthe pitch point.
 5. A gear set comprising a pair of mating gears havingconjugate gear teeth, the path of contact of said pair of gears being acurve substantially satisfying the differential equation: where: R1pitch radius of gear no. 1, R2 pitch radius of gear no. 2, phi p theinstantaneous value of the pressure angle at the pitch contact point, s,phi are the polar coordinates of the path of contact, referred to thepitch point, with phi also being the instantaneous pressure angle at anycontact point, Rho 1 radius of curvature of profile of gear no. 1, Rho 2radius of curvature of profile of gear no. 2, Rho r relative radius ofcurvature, and by definition: Kr 1/ Rho r 1/ Rho 1 + 1/ Rho
 2. 6. A gearset as defined in claim 5 wherein said path of contact corresponds tothe approximate solution to said differential equation which is definedby the equation: sin phi ao + a1 sigma + a22 where: sigma s/R1 ao sinphi p
 7. A gear cutting tool for cutting one of a set of mating gearteeth, the basic rack form of said cutting tool being derived from theequation: sin phi ao + a1 sigma + a2 sigma 2 where: sigma s/R1 ao sin pR1 pitch radius of gear no.
 1. R2 pitch radius of gear no.
 2. phi p theinstantaneous value of the pressure angle at the pitch contact point. s,phi are the polar coordinates of the path of contact, referred to thepitch point, and phi also being the instantaneous pressure angle at anycontact point.